How to make a circular cut?
Mesh LoopTools Click Circle

Mesh Loop Tools or Loops Tools is an Addon which must be installed in User Preferences.  That takes 88 seconds or less. 

Know that the faces of a mesh can be UV unwrapped and assigned a texture and painted. 

Simplified form mesh. Clearly the more low poly your mesh,  the more the hole will look like a polygon or octagon .. rather than a smooth circle with high vertex density.



Stages of editing left to right top to bottom.  Edit Mode.


Yellow vertex dense cylinder
Blue Partial quad strip extrusion
Yellow Loop Cuts for local vertex density
Blue Two Patches cut out in square on both sides.  Bridge Edge Loops





Yellow Mesh Loop Tool click Circle changes rectangle hole to circular hole. Select the vertices in edit more beforehand. See Side Panel. This is the step most relevant to the OP





Blue Proportional Editing for edge protrusion
Please improve this to suit you needs
Technique only. No artistry focus.